Problem 97 - Elementary Algebra

Question

Use the definition of percent to convert to fractions. $$ 3313 \% $$

Step-by-Step Solution

Answer

3313% is equivalent to the fraction \(\frac{3313}{100}\).

Step 1: Understanding Percent

A percent is a way of expressing a number as a fraction of 100. The symbol '%' means 'per hundred'. Thus, 3313% is equivalent to the fraction \( \frac{3313}{100} \).

Step 2: Writing as a Fraction

Using the understanding from the previous step, write the percent as a fraction: \[ 3313\% = \frac{3313}{100} \] This means 3313 percent is 3313 out of 100.

Step 3: Simplifying the Fraction

Check if \( \frac{3313}{100} \) can be simplified. Since 3313 and 100 have no common factors other than 1 (3313 being a prime number), the fraction is already in its simplest form.

Key Concepts

Fractions

Fractions are a fundamental way to express parts of a whole. When we talk about a fraction, we usually refer to two numbers separated by a slash, like this: \( \frac{a}{b} \). The top number is called the numerator, which shows how many parts we have, while the bottom number is called the denominator, which indicates the total number of equal parts in a whole.
Fractions can represent numbers less than one, equal to one, or greater than one. For example:

• \( \frac{1}{2} \) means one part out of two, or half.
• \( \frac{3}{3} \) means three parts out of three, which equals one whole.
• \( \frac{5}{2} \) means five halves, which is more than one whole.

Understanding fractions is crucial for many mathematical operations and concepts.

Converting Percentages to Fractions

Converting percentages to fractions is a straightforward process. A percentage is a way to express a number as a fraction out of 100. The word 'percent' literally means 'per hundred'. To change a percentage into a fraction:

• Write down the percentage number as the numerator.
• Place 100 as the denominator.
• Simplify the fraction if possible.

For example, if you start with 55%:

• Write it as \( \frac{55}{100} \).
• Then simplify by finding the greatest common divisor (GCD).
• Both numbers are divisible by 5, so \( \frac{55}{100} = \frac{11}{20} \).

However, with percentages like 3313%, you may find that the fraction is already in its simplest form if no GCD other than 1 exists between the numerator and 100.

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form. This involves dividing both the numerator and denominator by their greatest common divisor (GCD), which is the largest number that divides both without leaving a remainder.
Here’s a step-by-step process:

• Find the GCD of the numerator and the denominator.
• Divide both numerator and denominator by the GCD.
• The result is the fraction in simplest form.

For instance, to simplify \( \frac{40}{60} \):

• The GCD of 40 and 60 is 20.
• Divide both by 20 to get \( \frac{2}{3} \).

In cases where the fraction is already simplified, like \( \frac{3313}{100} \), it means no further reduction is possible since the numerator and denominator don't have a common factor other than 1.